How fast does heat travel through conduction, convection, and radiation?
You are looking for the rate of heat transfer $\dot q$ (joules per second).
Conduction
Fourier's law states the amount of heat conducted a certain distance/depth into a material per second, when there is a temperature difference:
$$\dot q=A \kappa \frac{dT}{dx}$$
- $A$ is contact area,
- $\kappa$ coefficient of thermal conduction (material specific),
- $T$ temperature and
- $x$ depth (distance the heat has "travelled").
The term $\frac{dT}{dx}$ is the so-called temperature gradiant (often written as $\nabla T$). For a symmetrical object, this is simplified to $\frac{\Delta T}{\Delta x}$.
Convection
Newton's law of cooling states the amount of heat transfered between the surface of a submerged object and the fluid every second:
$$\dot q=Ah(T_{s}-T_{\infty})$$
- $A$ is contact area (surface area if submerged),
- $T_s$ surface temperature of object,
- $T_\infty$ fluid temperature far away from (not affected by) the object and
- $h$ (often $h_c$) the convective heat transfer coefficient (material, fluid and process specific).
Convection is not simple. Convection can be forced (as by pumping) or natural (as by buoyancy), and the relative velocity $v$ of an object moving through a fluid, viscosity $\mu$ etc. play big roles. The equation appears simple at first sight, but includes the parameter $h$, which takes into account the process-specific details and may be very complicatedly tied upon other factors. This source shows some rough examples of values of $h$ in different situations.
In specific engineering cases, you either estimate (or numerically model) $h$ or apply a correlation of $h$ with other parameters that someone have found, if such exists for that specific case. Or you experiment and measure your way through it.
Radiation
The Stefan-Boltzmann law states that any surface with a temperature above $0\;\mathrm K$ radiates heat constantly:
$$\dot q=\varepsilon A\sigma T^4$$
- $A$ is exposed (uncovered) surface area,
- $\sigma$ Stefan's constant $\sigma=5.67\times 10^{-8}\;\mathrm{\frac{W}{m^2K^4}}$,
- $T$ temperature and
- $\varepsilon$ emissivity of the object (material and surface specific).
An emissivity of $\varepsilon=1$ gives an ideally radiating body, a so-called blackbody.
Note: Literature might often omit the $A$ in the three laws above. The laws are then stated as heat fluxes instead of heat rates. When dividing through with $A$ the left-hand-side becomes $\frac{\dot q}A$, called heat flux (sometimes given the symbol $\Phi$ or $\Phi_q$ or $\Phi_h$), which is simply heat transfered per square meter every second.
Does heat have a specific speed or does the speed depend on the type of material it's going through?
This is not an either-or case. Heat moves at a specific rate and that depends on the materials involved. See the few parameters in the description above, which depend on material.
From what I understand heat travels through conduction by 2 objects at different temperatures.
Heat only flows if there is a temperature difference, yes. The fact that you have two object is irrelevant - same is the case within one object. A temperature difference across two points will cause heat conduction, if they are in physical contact.
$cm\delta T=\delta Q$ , that's pretty much self evident , for the reference material the $\Delta Q$ is calculated in isolation , maybe with a burner or something , by heating it , Now you already have the thermometer and from the amount of fuel exhausted you know the heat change for say in the temperature change $\Delta T$ . Now if we have a same temperature change in your two body setting then you already know the heat change in one of the body from the above experiment .So in the second equation you've put $c_1m_1\delta T$ is known because its equal to $\delta Q$ , you see there is no variable except $c_2$ now.
How is the calorific value of fuel calculated ?
The fuel is put in a bomb calorimeter , upon burning the fuel releases heat and heats up the surrounding water , of which we calculate the temperature change.The unit of heat is defined with respect to water (calorie units) as you must be knowing.Since you know the temperature change , of water , you directly know the number of calories of heat released by fuel(calorific value) , problem solved , now you can proceed with the first paragraph.
Best Answer
From a biochemical point of view, heat detection is achieved by proteins at the surface of nerve cells. They basically just trigger a nerve signal above a given temperature. So they DO detect temperature and not a "heat flux". It may seem surprising that nerve cells react so quickly but the increase/decrease in temperature does not need to go all through the skin. It just need to be detected at the surface of the skin and a difference of 1°C is enough to start feeling a temperature change.
Interestingly, the temperature threshold can be changed by some well-known chemicals. For example, capsaicin (from hot peppers) will lower the temperature threshold for the heat-sensitive TRPV1 protein. This is what causes the burning sensation when eating spicy food. On the opposite, menthol (from mint) tricks the TRPM8 protein (and many others) into considering the temperature is lower than it actually is, which gives this sensation of cold in the mouth.
EDIT
The initial question took as an example the feeling we all know when we enter a cold room or touching objects. This edit is meant to address that.
It is true that, after some time, all the objects in a room will have the same temperature but our skin will not, just because our body produces heat and the air surrounding us is a poor conductor. If you take a thermometer in your fist, you should read roughly 27-29°C,(1) so let's consider that it is our skin temperature, for the sake of the demonstration. Also, I will consider that everything happens while temperature is 24°C max, and 22°C in the imaginary bathroom.
We can feel a big difference if we step on the bathroom tiles compared to what we feel if we take a wooden (or plastic) object in our hand. The tiles feel cold while the wooden object feels warm. A tile is usually a decent conductor of heat, so when we step on it, it rapidly cools down our foot sole and we feel it in less than a second. By contrast, a wooden (or plastic) object is an insulator. On contact with our skin, the exchange of heat is very slow, so the temperature of our skin will not change immediately and we interpret that as being a kind of neutral or warmish feeling.
(1) Of course it depends on many factors.